Why do Walter Rudin's proofs in real analysis often seem so elusive and clever?

[mix34]

Dear all,
I currently a student in mechanical engineering and i reached the conclusion that maths from the point of view of mathematicians is lot more interesting than the eyes of engineers (for me at least).

One of my friends in the maths department suggested to me to read real analysis Walter Rudin as a starting point and combine it with Burkill a first course in mathematical analysis.

In general i am just in the beginning of reading these books (just chapter 1) and i find both quite easy to read and understand each ones proof. However whenever i study Rudin and i read his proof i am not sure why his proof follows these steps (why use the argument the way he does? I can see why his proof are so well structured but how did he know to make this structure? Even the way he is phrasing and the sequence of arguments are deliberate towards the proof). I always try to prove the same theorem as him before reading his proof but i end up with an alternative method (most of the times quite different, though i still use the same theorems but not the same exact arguments or at best my phrasing is not the same does this mean my proof is wrong?). In general i always aim not to deviate beyond what is assumed in the chapter so to me it seems that my proof also always makes sense.

My questions are: what am i missing from Rudin and how can i make sure that my proof is right without doubts just like Rudin's (i was thinking about using propositional logic within my own proof but this could be too much)??
Am i on the right path with Rudin that makes me ask these questions? Or should i stick to Burkill, which i does not really make me have doubts (he is way more intuitive).

Thank you for your time and effort

 

[Number Nine]

"Little Rudin" is a classic, and is a good textbook for people who have a strong background in pure math, and are already familiar with the basics of real analysis. That said, I would absolutely not recommend it as an introduction. Rudin's proofs are very "nice", in the sense that they tend to be pretty and elegant, but they usually don't do a very good job of making the theorems clear. For intro real analysis, I'm a fan of Shilov's text

https://www.amazon.com/dp/0486689220/?tag=pfamazon01-20

though I'm sure someone here can give a better recommendation.

 

[MarneMath]

I agree with @Number Nine . Rubin's proofs tend to be clever, elegant and sometimes appear like they have little motivation. I had a very hard time understanding the proof and motivation. I spent my hours in my professor's office trying to understand it, and often times he would just go through a more straight forward proof that was more tedious to write but made more sense to me.Also it's not unusual to have to spent a week or so rereading a single chapter in Rubin. Many moons ago, I recall spending an entire day on a single page. There's a lot of subtleness in mathematical books. Often time catching the key phrase or word in a passage in the right light clears up the entire haze.